Editing Absolute magnitude (section)

=== Apparent magnitude ===
{{Main|Apparent magnitude}}

The Greek astronomer [[Hipparchus]] established a numerical scale to describe the brightness of each star appearing in the sky. The brightest stars in the sky were assigned an apparent magnitude {{math|1=''m'' = 1}}, and the dimmest stars visible to the naked eye are assigned {{math|1=''m'' = 6}}.<ref name="Carroll"/> The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitude {{mvar|M}} and apparent magnitude {{mvar|m}} from any distance {{mvar|d}} (in [[parsec]]s, with 1 pc = 3.2616 [[light-year]]s) are related by
<math display="block"> 100^{\frac{m-M}{5}}=\frac{F_{10}}{F} = \left(\frac{d}{10\;\mathrm{pc}}\right)^{2}, </math>
where {{mvar|F}} is the radiant flux measured at distance {{mvar|d}} (in parsecs), {{math|''F''<sub>10</sub>}} the radiant flux measured at distance {{math|10 pc}}. Using the [[common logarithm]], the equation can be written as
<math display="block"> M = m - 5 \log_{10}(d_\text{pc})+5 = m - 5 \left(\log_{10}d_\text{pc}-1\right),</math>
where it is assumed that [[Extinction (astronomy)|extinction from gas and dust]] is negligible. Typical extinction rates within the [[Milky Way]] galaxy are 1 to 2 magnitudes per kiloparsec, when [[Dark nebula|dark clouds]] are taken into account.<ref name="Unsoeld2013"/>

For objects at very large distances (outside the Milky Way) the luminosity distance {{math|''d''<sub>L</sub>}} (distance defined using luminosity measurements) must be used instead of {{mvar|d}}, because the [[Euclidean space|Euclidean]] approximation is invalid for distant objects. Instead, [[general relativity]] must be taken into account. Moreover, the [[cosmological redshift]] complicates the relationship between absolute and apparent magnitude, because the radiation observed was shifted into the red range of the spectrum. To compare the magnitudes of very distant objects with those of local objects, a [[K correction]] might have to be applied to the magnitudes of the distant objects.

The absolute magnitude {{mvar|M}} can also be written in terms of the apparent magnitude {{mvar|m}} and [[stellar parallax]] {{mvar|p}}:
<math display="block"> M = m + 5 \left(\log_{10}p+1\right),</math>
or using apparent magnitude {{mvar|m}} and [[distance modulus]] {{mvar|μ}}:
<math display="block"> M = m - \mu.</math>

==== Examples ====
[[Rigel]] has a visual magnitude {{math|''m''<sub>V</sub>}} of 0.12 and distance of about 860 light-years:
<math display="block">M_\mathrm{V} = 0.12 - 5 \left(\log_{10} \frac{860}{3.2616} - 1 \right) = -7.0.</math>

[[Vega]] has a parallax {{mvar|p}} of 0.129″, and an apparent magnitude {{math|''m''<sub>V</sub>}} of 0.03:
<math display="block">M_\mathrm{V} = 0.03 + 5 \left(\log_{10}{0.129} + 1\right) = +0.6.</math>

The [[Black Eye Galaxy]] has a visual magnitude {{math|''m''<sub>V</sub>}} of 9.36 and a distance modulus {{mvar|μ}} of 31.06:
<math display="block">M_\mathrm{V} = 9.36 - 31.06 = -21.7.</math>